Calculation of the Cohesive Energy of Metallic Iron

The cohesive energy of metallic iron is calculated for the bodycentered cubic structure in a singlet spin state at 0\ifmmode^\circ\else\textdegree\fi{}K. The potential field acting on each electron is taken to be that of the ion core and of the remaining valence electrons in the same lattice cell; thus the calculation becomes equivalent to one for the free atom as the lattice constant is increased. Tight-binding wave functions are used, but they are modified by expanding the contributions from neighboring atoms in a power series within a cell, and orthogonalizing to core states. Evaluating the complete wave function in each cell eliminates the need for multicenter integrals otherwise required in the tight-binding method. The wave functions for wave vectors in directions of high symmetry have a rather simple form, and can be described by a few parameters. States near the bottom of the $3d$ band tend to have a more diffuse charge distribution than do the states near the top of the band. Thus the x-ray scattering factor per electron for a partially filled $3d$ band will be less than that for a full band. Calculations of the energy of the solid are made for three values of the atomic sphere radius, ${r}_{s}$, using atomic wave functions from the $3{d}^{7}4s$ configuration. The indicated configuration in the solid is close to $3{d}^{7}4s$, making the calculation approximately self-consistent. The calculated width of the occupied portion of the $3d$ band is 0.33 ry. We find the cohesive energy of metallic iron to be 0.43\ifmmode\pm\else\textpm\fi{}0.2 ry per atom, which is consistent with the experimental value, 0.32 ry. The equilibrium lattice constant and the compressibility are both found to be in good agreement with experiment. An attempt to replace the Coulomb hole used in the main calculation by an exchange hole, using a single Slater determinant wave function, gave far too little binding.